By Earl A. Coddington

The prerequisite for the research of this booklet is an information of matrices and the necessities of features of a posh variable. it's been built from classes given via the authors and possibly includes extra fabric than will normally be lined in a one-year path. it really is was hoping that the publication might be an invaluable textual content within the software of differential equations in addition to for the natural mathematician.

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REVIEW PROBLEMS 11. v—>« x—>a is countable at most. 12. Let {r,,r 2 ,... } be an enumeration of all the rational numbers in the interval Define a function [ 0 , 1 ] and for each JCG[0, 1] let Ax = {nE:N:rn

Let X be a nonempty set. Then τ = {0, A"} is a topology on X, called the indiscrete topology. This topology is the smallest (with respect to inclusion) possible topology o n l . 3. Let I b e a nonempty set. Then r=<3)(Ar) is a topology on X. Here every subset of X is an open set. This topology is called the discrete topology, and it is the "largest" possible on X. 4. Let (X,d) be a metric space, with the set X uncountable. Let τ denote the collection of all subsets 0 of X such that for each x £ 0 there exists r > 0 and an at most countable subset A of X (both depending on x) such that x&A and B(x, r)~A C0.

Simmons, Introduction to Topology and Modern Analysis. New York: McGraw-Hill, 1963. CHAPTER 2 GENERAL TOPOLOGY AND FUNCTION SPACES The role of open and closed sets in metric spaces has been discussed previously. Now the fundamental notion of an open set will be generalized by introducing the concept of a topological space. The properties of open and closed sets will be studied in this setting. This chapter is devoted entirely to topological and function spaces and emphasizes the results needed for this book.